Some examples of first exit times

TL;DR

This paper computes the expected first exit times of Brownian motion from various domains in Euclidean and hyperbolic planes, providing insights into stochastic processes in different geometries.

Contribution

It introduces explicit calculations of first exit times for Brownian motion in diverse geometric domains, expanding understanding of stochastic behavior in Euclidean and hyperbolic spaces.

Findings

Explicit formulas for Euclidean domains

Expected exit times in hyperbolic geometry

Comparison of stochastic behaviors across geometries

Abstract

The purpose of this article is to compute the expected first exit times of Brownian motion from a variety of domains in the Euclidean plane and in the hyperbolic plane.